"Accessible microstates" is related to things like activation energy and metastability.
For example, think about a diamond sitting on a table at room temperature and atmospheric pressure. There are microstates for the diamond in which the carbon atoms are rearranged into a piece of graphite. But those microstates will essentially never occur no matter how long you wait and stare at this diamond sitting on the table. So the system has these microstates, where the diamond's atoms are rearranged into graphite, but these microstates are not "accessible" microstates under the circumstances. We need to ignore these microstates when computing the diamond's heat capacity etc.
It's not because the diamond is more stable (lower-energy), because it's equally true the other way around: A lump of graphite on a table will not spontaneously reorganize into diamond.
(For what it's worth, I think I heard somewhere that graphite, not diamond, is the most stable form of carbon at standard temperature and pressure.)